CHAPTER 4 : Convergence Time of Interdomain Routing
4.5 Convergence Time for Restricted Preferences: a Dichotomy Theorem
Although path linearization guarantees convergence after n phases of execution, this does not mean that the convergence time is polynomially bounded since there could be arbitrarily long sequences of improving moves during a phase. In this and the next sections, we study the convergence time of linearizable, but not linearized, networks (i.e., executing the protocol under partial preferences) over restricted families of preference systems.4 We start with the following families of preference systems.
Definition 4.14 (hs, li-Preference Systems). A preference L is a hs, li-preference system iff for the preference Lv of each node v, (i) there are at most spaths in |Lv|, i.e., |Lv| ≤s,
4
The execution model for partial preferences implicitly assumes that paths specified in the preference is better than the rest. By simply examining the resulting path digraph, it can be verified that having these additional preferences does not affect the linearizability of a network.
ak bk dk . . . a1 b1 d1 a0
ck c1
t
Bk B1
Figure 5: A network with constant preference size and constant-length paths, that takes an exponentially long time to converge.
and (ii) for each pathp in Lv, the length ofp is at most l, i.e., maxp∈Lv{|p|} ≤l.
We establish here the following dichotomy: even a linearizable network with only a h2,3i- preference system could encounter exponentially many improving moves before convergence, while any linearizable network with a h2,2i-preference system (resp. a h1,3i-preference system) always converges after at mostn2 (resp. 2n) improvements.
4.5.1. Exponential Convergence Time for h2,3i-Preferences
Our first result shows that there exists a family of (even acyclic) networks such that the preference of any node contains at most two paths where each path has length at most three, and yet convergence may take time exponential in the size of the network.
Theorem 4.3. For any k≥1, there exists a network NhG(V, E), L, ti, whereG is a DAG on n = 4k+ 2 nodes, and L is a h2,3i-preference system, s.t. the maximum convergence time ofN is 2Ω(n).
Note that G is a DAG immediately implies that the network is free of dispute wheel, and hence linearizable. A networkN satisfying the conditions of the above theorem is depicted in Figure 5, with the destination node t shown as the ‘ground’. The network consists of a special nodea0, followed byk=bn/4cblocksBi of four nodes each, namedai,bi,ci anddi.
The preferences for each node in a blockBi areLai ={aibidit ait},Lbi ={bicitbidit},
Lci ={ciditcit}and Ldi ={diai−1tdit}. The number of paths in the preference of for
each node is 2 and every path has at most 3 hops.
The central idea is a pattern of activations for the nodes in each blockBi, whereby a ‘flip’
state) triggers two flips for node ai. Accordingly, the sequence of blocks can be made to
amplify a single flip ata1 into exponentially many flips for the subsequentai.
For the interest of space, in the following, we will present the activation sequence of each block and explain how it leads to amplification, while a detailed proof of correctness is supplied in [68]. The order of each block is defined as follows.
Definition 4.15 (Block Order). For 1≤i≤k, let σi be the following ordering of nodes in Bi:
di, ai, bi, ai, ci, bi, ai, di, ci, ai. We will refer to this ordering as theblock order of Bi.
We consider the block order of activation starting with the following state.
Definition 4.16 (StateS1). For any 1≤i≤k, the S1 state of the block Bi is defined as S1(ai) = (ai, t), S1(bi) = (bi, di), S1(ci) = (ci, t), and S1(di) = (di, ai−1).
The activation of the block order onBi starting from the stateS1 is illustrated in Figure 6, where the red (or thick) edges represent the current state. The following lemma provides the essential ‘amplification’ step for the exponential time result. Recall that we say that a node ‘flips’ when it changes its assigned edge and then changes back. The lemma shows that a flip for node ai−1 can cause a double flip for node ai by activating the nodes in Bi
according to the block order (with the activation of ai−1 interposed). Therefore the node
ai+1 can be made to flip four times, and so on.
Lemma 4.5.1. For any 1 < i ≤ k, suppose that the block Bi is in state S1, the state of
ai−1 is (ai−1, bi−1), and the node ai−1 will shortly change to (ai−1, t). Then there exists a
schedule such that each node makes an improving move when activated, and the final state of Bi is still S1. Moreover, during activating under this schedule, the state of di flips once (from(di, ai−1) to(di, t) and then back), andai flips twice (from (ai, t) to(ai, bi)and back, twice).
ai bi di ai bi di ci ci t t Bi- State 1 Bi- State 2 ai bi di ai bi di ci ci t t Bi- State 3 Bi- State 4 ai bi di ai bi di ci ci t t Bi- State 5 Bi- State 6 ai bi di ai bi di ci ci t t Bi- State 7 Bi- State 8 ai bi di ai bi di ci ci t t Bi- State 9 Bi- State 10 ai bi di ci t Bi- State 11
Figure 6: States resulting from applying activation sequence σi to blockBi.
activation of ai−1 to adopt (ai−1, t) occurring part way through.
The progression of states leading to required effect is shown in Figure 6, and explained in detail in Table 1. Note that ai−1 will activate and adopt (ai−1, t) after state 2 but before state 9.
Consequently, the transition to state 2, where node di switches from an unranked path
beginning with (di, ai−1) to its second-best path dit is an improving move, and so is the
transition to state 9, whereai−1 has switched to (ai−1, t) and sodi can improve to its best
path, diai−1t.
State Activate Old edge New edge Improvement reason 1 di (di, ai−1) (di, t) ditdi diai−1bi−1. . . 2 ai (ai, t) (ai, bi) aibiditai ait 3 bi (bi, di) (bi, ci) bicitbi bidit 4 ai (ai, bi) (ai, t) aitai aibicit 5 ci (ci, t) (ci, di) ciditci cit 6 bi (bi, ci) (bi, di) biditbi bicidit 7 ai (ai, t) (ai, bi) aibiditai ait 8 di (di, t) (di, ai−1) diai−1tdi dit 9 ci (ci, di) (ci, t) citci cidiai−1. . . 10 ai (ai, bi) (ai, t) atai aibidiai−1. . .
Table 1: Sequence of improving moves for Lemma 4.5.1.
4.5.2. An Upper Bound on Convergence Time
We now investigate the maximum convergence time of linearizable networks. Firstly, note that any h1, li-preference system for any positive integer l, will converge after at most 2n
improvements since a node v can only switch from being not connected to t to choosing
some neighbor u where u has a path to t, or switch from choosing some neighbor u to
choosing the neighbor of the only path inv’s preference (when this path becomes available). Hence, we only need to establish maximum convergence time forh2,2i-preference systems, and combining with our construction for showing that h2,3i-preference systems lead to exponentially many improvements (Theorem 4.3), we have a dichotomy theorem. In fact, we will establish the maximum convergence time for networks withh2, li-preference systems for any positive integerl, which, as a special case, implies poly-time convergence forh2,2i- preference systems.
We define the function L(l, n) for any positive integers l and n, to be the sum of a list of integersL(l, n) =Pn
i=1ai, wherea1= 0 andai = (l−1)ai−1+ 2 for alli, and establish the following theorem.
Theorem 4.4. For any positive integer l, for any linearizable network NhG(V, E), L, ti
whereLis ah2, li-preference system, the maximum convergence time ofN is at mostL(l, n), where n is the number of nodes inG.
We first establish that the total number of improvements made by any node v is upper
bounded by the total number of improvements made by the nodes on the best path of v
(Lemma 4.5.2). To utilize this fact, we introduce a process for finding a sequence of the nodes such that every nodev appears after any other node on the best path of v (Lemma 4.5.3). Examining the nodes under this sequence, we show that for any i, the i-th node in the sequence can make at mostai improvements, hence proving thatL(l, n) is an upper bound
of the total number of improvements.
We denote by Imp(v) the maximum number of improvements that the node v can make.
(The proofs of Lemma 4.5.2, 4.5.3 are deferred to [68].)
Lemma 4.5.2. For any node v in a h2, li-preference system network whose best path is denoted by vu1u2. . . ujt, Imp(v)≤Pji=1Imp(ui) + 2.
We use the notion of “good” node to define/find a sequence of the nodes when examining their number of improvements.
Lemma 4.5.3. For any linearizable network hG(V, E), L, ti, for any T ( V with t ∈ T,
there exists at least one nodev∈V\T s.t. eitherLv =∅, or for the best path ofv vu1. . . ujt, ui ∈T for all i. We call such a node v a good node forT.
Proof of Theorem 4.4. We will create a list of subsets ofV,{Ai}ni=1 by starting fromA1=
{t}, adding one node every step, and ending withAn=V. Letai = max{Imp(v)|v∈Ai}.
We will prove in the following that ai ≤(l−1)ai−1+ 2. Since the node added at thei-th
step can make at most ai improvements, by summing up the number of improvements of
all nodes, we achieve the upper bound ofL(l, n) total improvements.
Starting fromA1 ={t}, we now show the transition fromAi−1toAi, i.e., how to pick a node vthat formsAi=Ai−1∪{v}withai ≤(l−1)ai−1+2. By Lemma 4.5.3 withT =Ai−1, there exists a node v with either empty preference, or for v’s best path, vu1u2. . . ujt, all nodes
except v belong toAi−1. We pick any such nodev (i.e., any good node forAi−1). For the first case,vhas an empty preference implies that the only improvement vcan make is from
⊥to one of its neighbor. For the second case, by Lemma 4.5.2,Imp(v)≤Pj
i=1Imp(ui)+2.
Sincej≤(l−1) and alluibelongs toAi−1, we haveImp(v)≤Pji=1ai−1+2≤(l−1)ai−1+2. Since ai = max{ai−1, Imp(v)},ai ≤(l−1)ai−1+ 2.
A simple calculation shows that L(2, n) ≤ n2, and for any l ≥ 3, L(l, n) ≤ 2(l −1)n.
Therefore, in particular, for any network with a h2,2i-preference system, the maximum convergence time is at mostn2. For a network withh2,3i-preference system, the maximum
convergence time is at most 2n+1. Combined with Theorem 4.3, which shows that there
exist networks with h2,3i-preferences that take 2Ω(n) time to converge, we establish that theconvergence time complexity of networks withh2,3i-preference systems is 2Θ(n).